$\newcommand\R{\mathbb R}$This is not true in general. E.g., let $p(s):=(\arctan s,0,\dots,0)$. Then 
$$\bigcup_{s\in\R}N_s=(-\pi/2,\pi/2)\times\R^{n-1}\ne\R^n.$$ 

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After the above answer was posted, the OP has changed the question by adding the condition that $p(s)$ be pointwise polynomial, thus invalidating the above answer. However, this does not help: let 
$p(s):=(s_+^2,(1-s)_+^2,0,\dots,0)$, where $u_+:=\max(0,u)$. Then 
$$\bigcup_{s\in\R}N_s=[0,\infty)^2\times\R^{n-2}\ne\R^n$$
if $n>2$.